Silently, invisibly, ceaselessly, our planet Earth is showered by radio waves from every direction and from every region of space. This radio energy originates in our solar system, throughout the Milky Way galaxy, and far beyond, out to the remotest reaches of the universe. Detecting and unraveling the origins of these invisible signals is what radio astronomy is all about. This ever-present radiation provides astronomers with an alternate, non-optical window to the universe, revealing exotic and unfamiliar phenomena previously undetected by even the most powerful optical telescopes. For physics teachers, a classroom discussion of these radio discoveries, however brief, offers an opportunity for igniting interest (and possibly a career option) in even the most apathetic of students. This paper describes, first, the background of some of these events, and second (in the appendixes), a selection of numerical problems so that students can derive for themselves the truly mind-stretching features of these celestial objects.